Journal of Applied Mathematics

Volume 2012 (2012), Article ID 607105, 17 pages

doi:10.1155/2012/607105

Research Article

Periodic Solution of Prey-Predator Model with Beddington-DeAngelis Functional Response and Impulsive State Feedback Control

Chunjin Wei¹ and Lansun Chen²

¹Science College, Jimei University, Fujian, Xiamen 361021, China

²Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China

Received 18 June 2012; Revised 31 August 2012; Accepted 2 September 2012

Academic Editor: Shiping Lu

Copyright © 2012 Chunjin Wei and Lansun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A prey-predator model with Beddington-DeAngelis functional response and impulsive state feedback control is investigated. We obtain the sufficient conditions of the global asymptotical stability of the system without impulsive effects. By using the geometry theory of semicontinuous dynamic system and the method of successor function, we obtain the system with impulsive effects that has an order one periodic solution, and sufficient conditions for existence and stability of order one periodic solution are also obtained. Finally, numerical simulations are performed to illustrate our main results.

1. Introduction

The study of the dynamics of prey-predator system is one of the dominant subjects in both ecology and mathematical ecology due to the fact that predator-prey interaction is the fundamental structure in population dynamics. Many scholars have carried out the study of prey-predator system with various functional responses, such as Monod-type and Holling-type. It is well known that Beddington-DeAngelis functional response which was introduced by Beddington and DeAngelis et al. [1, 2] can avoid some of the singular behavior of ratio-dependent models at low densities and provide better description of predator feeding over a range of prey-predator abundances. Therefore, in this paper, we investigate prey-predator system with Beddington-DeAngelis functional response.

Impulsive differential equations have been widely used in various fields of applied sciences, for example, physics, ecology, and pest control. The majority of them just concern the system with impulses at fixed times [3–7]. However, in practical ecological system, the control measures (by poisoning or releasing the natural nenmy, etc.) are taken only when the amount of species reaches a threshold value, rather than the usual impulsive fixed-time control strategy. Impulsive state feedback equation is a powerful tool to manage these problems. Therefore, some researchers proposed the impulsive state feedback control models for population management [8–11].

Motivated by the above works, in this paper, we consider the following predator-prey system with Beddington-DeAngelis functional respose:

where , , , , , , , , and   are positive constants and represent the population density of prey (pest) and predator (natural enemy) at time , respectively. is the intrinsic growth rate and is the carrying capacity in the absence of predation. The predator consumes the prey with functional response of Beddington-DeAngelis type and contributes to its growth with rate . is the death rate of predator. , when the amount of the prey reaches the threshold at time , controlling measures are taken and the amount of the prey and predator abruptly turn to and , respectively. The functional response in system (1.1) is similar to the well-known Holling type II with an extra term in the denominator which models the mutual interference among predators. It also has some of the same qualitative behaviours as the classical ratio-dependent model (i.e., ), but is free from the singular behaviors of ratio-dependent model at low densities which is, in fact, the source of controversy [12–14]. For simplicity, let , then system (1.1) becomes the form:

We assume that , otherwise , we have as , which means that the predator population will die out, therefore, in the following, we always assume that .

We nondimensionalize system (1.2) with the following scaling: , , and have

where the nondimensional parameters are defined as , , , , , , , and . Obviously that , , , .

In this paper, we mainly discuss the existence and stability of periodic solution of system (1.3) by using the geometry theory of semicontinuous dynamic system and the method of successor function which were introduced in the paper [15, 16], as far as we know, there are few papers to apply these methods to prove the existence of order one periodic solution, which makes the study simpler and clearer. What is worth saying is that modeling thoughts and mathematical methods used in this paper are of more important theoretical and practical value.

An outline of this paper is as follows: some definitions and theorems are given for the later use in the next section. The qualitative analysis of the system without impulsive effects is given in Section 3. In Section 4, the existence and stability of order one periodic solution of system (1.3) are investigated. Numerical simulations and some discussions are provided in Section 5.

2. Definitions and Lemmas

Definition 2.1 (see [15]). Differential equation with impulsive state feedback control

where and are lines or curves on the plane . is called pulse set and is called phase set. The dynamical system which is constituted by the solution map of system (2.1) is called semicontinuous dynamical system which is denoted by . We assume that the map with the initial point is not in the pulse set , that is, , is a continuous map, , and is called pulse map.

In this paper, , . , , , then system (1.3) constitutes a semicontinuous dynamical system .

For any , the function defined as is continuous and we call the trajectory passing through point . The set is called positive semitrajectory of point . The set is called the negative semitrajectory of point . For the convenience, if , is defined as the first intersection point of and , that is, there exists a such that , and for .

Definition 2.2 (see [15]). A trajectory is called order one periodic solution with period if there exists a point and such that and .

Next we will give the definition of the successor function of semicontinuous dynamical system (1.3). First, we define a new number axis in set . On straight line , take the origin at point of coordinate axis and define positive direction and unit length to be consistent with coordinate axis , then we obtain a number axis . For any , let be coordinate of point .

Definition 2.3. Suppose be a map. For any , there exists a such that , , then is called the successor function of point , and the point is called the successor point of .

Remark 2.4. If , the trajectory with initial point is an order one periodic solution of system (1.3).

According to the continuity of compound function, we know the following.

Lemma 2.5. The successor function is continuous.

In system (1.3), the isocline is denoted by , the isocline is denoted by . Let be the intersection point of isocline and set . If there exist two points , , which are both below or above , satisfying , by the zero point theorem of continuous function in the closed interval, we know there exists a point which is between and such that , so we have the following.

Lemma 2.6 (existence theorem of order one periodic solution). The system (1.3) exists an order one periodic solution if there exist two points , , which are both below or above , satisfying .

3. Qualitative Analysis of System (1.3) without Impulsive Effects

In this section, we will study the qualitative characteristic of system (1.3) without impulsive effects. If no impulsive effects is introduced, then system (1.3) is

The equilibrium of system (3.1) satisfies

It can be seen that system (3.1) exist boundary equilibrium , , where satisfied and a positive equilibrium provided that , where , . In the following, we always suppose that .

Lemma 3.1. The system (3.1) is uniformly bounded.

Proof. Let the straight line , we have , then the trajectory of system (3.1) from the right of through the into the left. Define a function , where . The function intersects the line and axis at the points and , respectively. We have . we can choose enough large such that . Hence, the system (3.1) is uniformly bounded (see Figure 1). The proof is completed.

Figure 1: The uniformly bounded region.

In the following, we will analysis the stability of equilibrium and of system (3.1). Clearly, is saddle point and if , is stable, otherwise is saddle point and the positive equilibrium point exists. For the positive equilibrium point, we have the following result.

Theorem 3.2. The positive equilibrium point is globally asymptotically stable if .

Proof. The Jacobian matrix of system (3.1) at takes the form of

The eigenvalue problem for the provides the characteristic equation

where the coefficients , are

Note that , , then both of the eigenvalues have negative real part, we have that is locally asymptotically stable.

Let , and , where , , then we have

the system (3.1) has no closed trajectory, therefore is globally asymptotically stable. This completes the proof.

4. Existence and Stability of Periodic Solutions of System (1.3)

4.1. Existence of Order One Periodic Solution

In this section, we will investigate the existence of order one periodic solution of system (1.3) by using the method of successor function. Note that system (1.3) is a semicontinuous dynamical system, for convenience, any point , let denote its abscissa and be its ordinate. If , then impulse occurs at point , the impulsive function transfers the point into . Without loss of generality, unless otherwise specified we assume the initial point of the trajectory lies in . From discussion of Section 3, we know that is globally asymptotically stable when . If , , all the solutions of system (1.3) tend to the equilibrium after finite impulse, so we mainly pay attention to the case or and .

Case I (). In this case, sets and are both in the left side of point . Trajectory passing through point tangents to at point . Set , then impulse occurs at point . Suppose point is subject to impulsive effects to point , here , the position of has the following three cases.

Case ( ). coincides with , and the successor function of is , so trajectory and segment formulate an order one periodic solution of system (1.3).

Case ( ). In this case, is above . Set , in view of vector field and disjointness of any two trajectories, we know , so we have , then the successor function of is .

Take another point above A, where is small enough. Set , in view of continuous dependence of the solution on initial value and time, we know and the point is close to enough, so we have and the point is close to enough, since , then we obtain . By Lemma 2.6, we know there exists an order one periodic solution of system (1.3), whose initial point is between and in set (see Figure 2).

Figure 2: Illustration of system (1.3) for the case , .

Case ( ). In this case, is below . We have , so , then the successor function of is .

Take another point satisfying . Set , then , so we have . By Lemma 2.6, we know there exists an order one periodic solution of system (1.3), whose initial point is between and in set (see Figure 3). Therefore, we have the following theorem.

Figure 3: Illustration of system (1.3) for the case , .

Theorem 4.1. Suppose that , , then system (1.3) has an order one periodic solution.

Case II (). In this case, set is in the right side of and set in the left of . Denote the intersection point of isocline and set by . The trajectory passing through point tangents to at point . is defined as the first intersection point of and , that is, there exists a such that , and for , .

Case ( ). There exist such that , , and for . Suppose point is subject to impulsive effects to point .

If coincides with or , trajectory and segment or trajectory and segment formulate an order one periodic solution of system (1.3).

If is below , that is, , the successor function of is . Take another point satisfying . Set , then , then we have . We conclude that there exists an order one periodic solution of system (1.3), whose initial point is between and in set (see Figure 4).

Figure 4: Illustration of system (1.3) for the case , , , .

If is above , that is, , the successor function of is . Set , in view of the vector fields of system (1.3), we know , so we have . We conclude that there exists an order one periodic solution of system (1.3), whose initial point is between and in set (see Figure 5).

Figure 5: Illustration of system (1.3) for the case , , , .

If is between and , in view of vector fields, we know the trajectory of system (1.3) initiating any point between and of set will be free from impulsive effects and ultimately tend to stable point , thus there does not exist order one periodic solution in system (1.3) in this case. So we have the following.

Theorem 4.2. Suppose that , and , or , then system (1.3) has an order one periodic solution.

Case ( ). Denote the intersection point of isocline and set by . The trajectory passing through point tangents to at point . Set , then impulse occurs at point . Suppose point is subject to impulsive effects to point . Like the analysis of Case I, we can prove that there exists an order one periodic solution in system (1.3) in this case (see Figure 6).

Figure 6: Illustration of system (1.3) for the case , , .

Theorem 4.3. Suppose that and , then system (1.3) exists an order one periodic solution.

In the following, we analyze the stability of order one periodic solution in system (1.3). Firstly, we give one lemma to discuss the stability of this periodic solution of system (1.3).

Lemma 4.4. The -periodic solution of the system,

is orbitally asymptotically stable if the Floquet multiplier satisfies the condition , where

with

and , , , , , , , and are calculated at the point , , . is a sufficiently smooth function with grad , and is the time of the th jump.

The proof of this lemma is referred to Simeonov and Baĭnov [17].

In the following, we suppose this periodic solution of system (1.3) with period passes through the points , and (see Figures 2–6). As the expression and the period of this solution are unknown, we discuss the stability of this positive periodic solution by Lemma 4.4. In our case,

Then

Set

if are period function with , then

hence

Suppose , then

Because is period solution of system (1.3), we have . Obviously, if

Theorem 4.5. If system (1.3) has an order one periodic solution, this order one periodic solution is orbitally asymptotically stable provided

5. Numerical Analysis and Discussion

In order to verify the theoretical results in this paper, we consider the following example

In numerical simulation, let , , , , , and , by calculating we know , , . If , , , , then the time series and phase portrait can be seen in Figure 7, by analysis of Section 4, we know that in this case, system (1.3) has no periodic solution and all the solutions tend to the equilibrium after finite impulses. If , by Theorem 4.1, we know that system (1.3) has an order one periodic solution which is shown in Figure 8. As shown in Figure 9, if , , by Theorems 4.2 and 4.3, we know system (1.3) also has an order one periodic solution. The results show that prey-predator model with Beddington-DeAnglis functional response and impulsive state feedback control tends to stable state or be periodic, which means that the amount of prey is under control.

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(b)

(c)

Figure 7: Time series and portrait phase of system (1.3) with , , , , , and .

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Figure 8: Time series and portrait phase of system (1.3) with , , , , and .

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Figure 9: Time series and portrait phase of system (1.3) with , , , , and .

Choosing as a control parameter, the phase portraits of system (1.3) are shown in Figure 10 with different values , which not only indicate that the prey can be suppressed below certain level by using the impulsive state feedback strategy when the amount of the prey reaches the economic threshold , but also indicate that it is more easy to control the amount of the prey below certain economic threshold as increasing. On the other hand, if choosing as a control parameter, we can see that more numbers of impulses are needed to make the system tend to a stable periodic solution, that is, it is more difficult to control the amount of the prey below economic threshold as is decreasing (see Figure 11). Figure 12 shows that it is more easy to control the amount of the prey under certain economic threshold as is increasing or is increasing (see Figure 13).

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Figure 10: The phase portraits of system (1.3) with , , , , , and different : (a) ; (b) ; (c) .

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Figure 11: The phase portraits of system (1.3) with , , , , , and different : (a) ; (b) ; (c) .

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Figure 12: The phase portraits of system (1.3) with , , , , and different : (a) ; (b) ; (c) .

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Figure 13: The phase portraits of system (1.3) with , , , , and different : (a) ; (b) ; (c) .

According to the above analysis, it is obvious that the prey can be well suppressed below certain level by using impulsive state feedback strategy for the fact that the system has stable periodic solution under some conditions. The key to the system with impulsive state feedback control is to give the suitable feedback state (the value of ) and the control parameters (, and ) according to practice.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11171284), the Fujian Provincial Natural Science Foundation of China (2012J01012), the Fujian Provincial Education Fundation (JA12198), and the Scientific Research Foundation of Jimei University of China (ZC2011003).

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